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Thank you both for explaining the issues. Might it also help if the actual (often bell shaped) likelihood distribution was always displayed diagrammatically? The baseline could be marked with the 95% and 99% ‘limits’ (and labelled ‘uncertainty’ or ‘compatibility’). The position of the null hypothesis could also be marked on the baseline. If this first null hypothesis is placed -z SEMs from the observed mean x̅, another mirror null hypothesis can be placed +z SEMs from the observed mean x̅. This pair of null hypotheses would form the basis of a 2 sided P value.
The bell’s heights at each point represent the likelihoods of observing the study mean conditional all the possible hypothetical values of the ‘true’ mean IF (1) the study methods had been described impeccably AND (2) the distribution model (e.g. Gaussian) was appropriate AND (3) there was no prior information available about the distribution of likelihoods. The way the diagram is interpreted would depend on the reader’s approach (e.g. from a Frequentist point of view). However, based on the above three assumptions, the provisional Bayesian probability of observing a true result less extreme than the null hypothesis in the direction of the observed mean would be equal to the one-sided P value [1].
If other prior information were available then a Bayesian calculation would combine the prior distribution with the observed distribution. The mean and the updated combined distribution would be shifted up or down and the ‘spread’ or variance would be reduced.
Reference
1. Llewelyn H (2019) Replacing P-values with frequentist posterior probabilities of replication—When possible parameter values must have uniform marginal prior probabilities. PLoS ONE 14(2): e0212302. https://doi.org/10.1371/journal.pone.0212302
Competing interests:
No competing interests
12 September 2019
Huw Llewelyn
Retired general physician and endocrinologist, Honorary fellow in mathematics
Re: Are confidence intervals better termed “uncertainty intervals”?
Thank you both for explaining the issues. Might it also help if the actual (often bell shaped) likelihood distribution was always displayed diagrammatically? The baseline could be marked with the 95% and 99% ‘limits’ (and labelled ‘uncertainty’ or ‘compatibility’). The position of the null hypothesis could also be marked on the baseline. If this first null hypothesis is placed -z SEMs from the observed mean x̅, another mirror null hypothesis can be placed +z SEMs from the observed mean x̅. This pair of null hypotheses would form the basis of a 2 sided P value.
The bell’s heights at each point represent the likelihoods of observing the study mean conditional all the possible hypothetical values of the ‘true’ mean IF (1) the study methods had been described impeccably AND (2) the distribution model (e.g. Gaussian) was appropriate AND (3) there was no prior information available about the distribution of likelihoods. The way the diagram is interpreted would depend on the reader’s approach (e.g. from a Frequentist point of view). However, based on the above three assumptions, the provisional Bayesian probability of observing a true result less extreme than the null hypothesis in the direction of the observed mean would be equal to the one-sided P value [1].
If other prior information were available then a Bayesian calculation would combine the prior distribution with the observed distribution. The mean and the updated combined distribution would be shifted up or down and the ‘spread’ or variance would be reduced.
Reference
1. Llewelyn H (2019) Replacing P-values with frequentist posterior probabilities of replication—When possible parameter values must have uniform marginal prior probabilities. PLoS ONE 14(2): e0212302. https://doi.org/10.1371/journal.pone.0212302
Competing interests: No competing interests